1. Universal Semantic Communication Brendan Juba (Harvard and MIT) with Madhu Sudan (MSR and MIT) & Oded Goldreich (Weizmann)

4. 11010 0 HOW DO WE DEFINE THE “MEANING OF THE COMMUNICTATION? ??” TO BE CONTINUED…

5. MAN, WHAT THE EFF?? A FAILURE TO COMMUNICA TE!

6. I.Model of communication II.Theory of finite communication III.Example: computation IV.Model for infinite communication Outline

7. “Meaning” = Usage ENVIRONME NT =

8. Printer Printing, formally Printer driver Printer firmware ENVIRONME NT INTERFACE FIXED IN ADVANCE! GOAL OF COMMUNICATION

9. “USE R” “SERVE R” ENVIRONMEN T BEHAVIOR DEFINED WITH GOAL Abstract goals of communication “G = (ENV,R)” FINITE GOAL OF COMMUNICATION: “USER ACHIEVES GOAL” IF USER “HALTS” WHEN R = 1 R:  {0,1} environment internal state σu2σu2 σu1σu1 σs2σs2 σs1σs1 U: Ω u × {0,1} *  Ω u × {0,1} * dist. over S: Ω s × {0,1} *  Ω s × {0,1} * dist. over

10. Goal of computation (function f) ENVIRONMEN T x x f(x) R = “user message = f(x)?”

11. 1.Goal of Communication 1.Universal user 2.Sensing function 3.Helpful server Key Concepts

12. Bob’s problem ? ? I DON’T KNOW WHICH ONE! P BOB WANTS TO PRINT SUCCESSFULLY, REGARDLESS OF WHICH PRINTER HE IS USING

14. Universal user NOTE: WE SHOULD SUCCEED FROM ANY STATE ENVIRONMEN T P -Universal user for printing P

15. ENVIRONME NT 11 01 ENVIRONME NT 11 01 I’M THROUGH WITH YOU THAT’S ALL I NEEDED TO HEAR! FROM ANY STATE?? I SURE BLEW THAT…

16. Summary: universal user Definition. A universal user for a goal G = (ENV,R) and a class of servers S is a user strategy s.t. for every server S in S and every initial state of S and ENV, the user achieves G. That is, halts when R = 1 (w.h.p.) WE WILL SAY THAT THE UNIVERSAL USER IS “EFFICIENT” IF, WITH EACH SERVER S IN S, THE USER RUNS IN SOME POLYNOMIAL TIME DEPENDING ON S, WITH THE GOAL- SPECIFIC SIZE PARAMETER DEPENDING ON ENV.

17. I.Model of communication II.Theory of finite communication III.Example: computation IV.Model for infinite communication Outline

18. IT’S ALL ABOUT THE FEEDBACK!!

19. 1.Goal of Communication 1.Universal user 2.Sensing function 3.Helpful server Key Concepts

20. ENVIRONMEN T I CAN STOP! Sensing functions: “safety” SENSING FUNCTION: V : user’s view  {0,1} “ V IS SAFE”: V = 1  R = 1 (w.h.p.) RECALL, REFEREE: R : environment’s view  {0,1}

21. Sensing functions: “viability” ENVIRONMEN T I CAN STOP! “ V IS VIABLE” IF THERE EXISTS SOME USER STRATEGY THAT RELIABLY OBTAINS V = 1

22. Theorem 1. If there is an efficiently computable S -safe and S -viable sensing function for a goal, then there is an efficient S -Universal user for that goal. ENUMERATE ALL USER ALGORITHMS, RUN EACH WITH CONSTANT FACTOR OVERHEAD: SAFE & VIABLE SENSING FUNCTION INDICATES WHEN TO HALT Achieving Universal Communication Each algorithm of length l gets ≈ 1/ l 2 2 l - share of the total running time

23. Theorem 2. There is a natural class of 2 l servers S s.t. a S -Universal user for any goal that requires the server to act experiences an overhead of Ω(2 l ) rounds. IT TAKES ≈ 2 l ROUNDS TO SEND ALL 2 l PASSWORDS OF LENGTH l ! NOTE: QUALITATIVELY OPTIMAL IN TERMS OF PROGRAM LENGTHS! Theorem 2. There is a natural class of 2 l servers S s.t. a S -Universal user for any goal that requires the server to act experiences an overhead of Ω(2 l ) rounds. Might still consider restricted classes where we can be efficient…

24. So what is Theorem 1 good for?? CHARACTERIZATION IN TERMS OF SENSING FUNCTIONS CAN BE USEFUL

25. Helpful servers ENVIRONMEN T “ S IS HELPFUL” IF THERE EXISTS SOME USER STRATEGY THAT RELIABLY SUCCEEDS AT G KEY DEF. #4…

26. SGSG

27. S G -Universal user for G ENVIRONMEN T SGSG N O C OMMON K NOWLEDGE N ECESSARY !

28. Theorem 3. If there is an efficient S -Universal user for a goal, then there is an efficiently computable S -safe and S -viable sensing function for that goal. THE FUNCTION THAT TELLS A UNIVERSAL USER WHEN TO HALT IS A SAFE & VIABLE SENSING FUNCTION

29. Main Theorem. There is an efficient S -Universal user for a goal if and only if there is an efficiently computable S -safe and S -viable sensing function for the goal. MORAL: SAFE & VIABLE SENSING FUNCTIONS ARE PRECISELY THE FUNCTIONS THAT TELL UNIVERSAL USERS WHEN TO HALT!

30. Theorem 4. If a sensing function is S G -safe for a goal G, then it is safe for G with all servers, even malicious and unhelpful ones. CAN CONSTRUCT A HELPFUL SERVER THAT BREAKS SAFETY WHENEVER SOME ADVERSARY CAN

31. SGSG SGSG Proof sketch: Theorem 4 ENVIRONMEN T I CAN STOP! NOT S G -SAFE FOR G

32. RECAP: 1. Sensing is necessary and sufficient 2. Sensing with helpful servers must also be safe with all servers We’ll see a more concrete interpretation of these theorems next…

33. I.Model of communication II.Theory of finite communication III.Example: computation IV.Model for infinite communication Outline

34. Goal of computation (function f) ENVIRONMEN T x x f(x) R = “user message = f(x)?”

35. For which problems can solutions be communicated without common knowledge?

36. S Competitive Proof Systems (Bellare-Goldwasser ‘94) “x  S” SOUNDNE SS (STANDAR D) PROVE IT! YOU AREN’T FOOLING ANYONE! COMPLETENESS (“COMPETITIVE PROVER”) WELL, I’M CONVINCED! EFFICIENT, GIVEN ORACLE FOR S

37. Theorem 5. Let G be the goal of deciding membership in a set S. Then there is a S G -universal user for G iff there are competitive proof systems for both S and S c. Corollary. If there is a S G -universal user for G then S is in PSPACE.

38. ENVIRONMEN T S Theorem 5: obtaining a competitive proof system from a universal user SGSG SGSG x x S(x) “x  S” NOT FOOLED: THEOREMS 3&4 TIME’S UP…

39. Theorem 5: obtaining a universal user from a competitive proof system S “x  S” x x HELPFUL SERVER I WON’T BE FOOLED!

40. Computational problems with universal users Any PSPACE-complete problem [Shamir’92] Any #P-complete problem [LFKN’92] Graph Isomorphism [GMW’91] Total functions in NP (solvable by Levin’s universal search algorithm [Levin’73]) – Integer Factoring – Discrete Logarithm – many more…

41. I.Model of communication II.Theory of finite communication III.Example: computation IV.Model for infinite communication Outline

42. REPEATING FINITE COMMUNICATION STRATEGY: PROBABILITY p OF FAILURE EACH SESSION… Multi-session goals EN V SESSION 1 … SESSION 2 SESSION 3 INFINITE SESSION STRATEGY: ZERO ERRORS AFTER FINITE NUMBER OF ROUNDS

43. Sensing for infinite goals SESSION 1 … SESSION 2 SESSION 3 EN V I’D BETTER TRY SOMETHING ELSE!! SAFETY: ERRORS DETECTED WITHIN FINITE # OF ROUNDS VIABILITY: FAILURES CEASE WITHIN FINITE # OF ROUNDS FOR AN APPROPRIATE COMMUNICATION STRATEGY

44. This weaker version of sensing suffices to construct universal users for infinite goals. But is it necessary??

45. 110 0110 11110 An impossible finite goal ENVIRONMEN T I WONDER IF IT PRINTED… RECALL: WE SHOULD STOP IN FINITE TIME

46. 110 0110 11110 A possible infinite goal ENVIRONMEN T PASSWORD FOUND IN FINITE # OF ROUNDS MORAL: FEEDBACK IS UNNECESSARY!

47. We saw a model for capturing problems of misunderstanding in communications systems. We also saw some limits of “strong” solutions to this problem.

48. THERE EXISTS SOME USER STRATEGY THAT RELIABLY SUCCEEDS AT G 1.Goal of Communication 1.Helpful server 2.Universal user 3.Sensing function Key Concepts G = (ENV,R:  {0,1}) environment internal state FOR EVERY SERVER S IN S AND EVERY INITIAL STATE OF S AND ENV, THE USER ACHIEVES G V : user’s view  {0,1} SAFETY: ERRORS DETECTED WITHIN FINITE # OF ROUNDS SAFETY: V = 1  R = 1 VIABILITY: FAILURES CEASE WITHIN FINITE # OF ROUNDS FOR AN APPROPRIATE COMMUNICATION STRATEGY VIABILITY: THERE EXISTS SOME USER STRATEGY THAT RELIABLY OBTAINS V = 1

49. “Meaning” is relevant I DON’T THINK SO. RIDICULOUS! I ONLY NEED MY PRINTER TO RECEIVE THE SAME BITS I SENT, RIGHT?? IF ONLY, BOB… 11010 0

50. Choose your own counterexample Map is infinite Printer is black-box Different messages may have same effect Verifiable? NOTE: NOT A “REAL” PROBLEM!

51. 110 0110 11110 Password-protected servers ENVIRONMEN T 11110

52. 0001 111111 00110 11110 1100001 100111 0 0001000 001 11 PW ( )

53. Theorem 2. A PW( S ) -Universal user for a goal that requires the server to act must run for Ω(2 l ) rounds with servers with passwords of length l. IT TAKES ≈ 2 l ROUNDS TO SEND ALL 2 l PASSWORDS OF LENGTH l ! NOTE: QUALITATIVELY OPTIMAL IN TERMS OF PROGRAM LENGTHS!

54. FINITE vs. INFINITE Strong short-term guarantee Strong sensing necessary NO short term guarantee Strong long-term guarantee Sensing seems unnecessary

55. Open problem Find an “interesting” class of servers for which a universal user can be more efficient than a trial-and-error search. IDEALLY, “BOUNDED-OPTIMAL” SEARCH…

56. 1.Goal of Communication 1.Helpful server 2.Universal user 3.Sensing function